a) Expand and simplify: $$ \begin{array}{llll} ext { (1) } & (x+7)(x-7) & ext { (2) } & (x-3)(x+3) \ ext { (4) }(9 x+4)(9 x-4) & ext { (5) } & (3 x-2 y)(3 x+2 y) & (2 x-1)(2 x+1) \ \begin{array}{ll} ext { (7) } & \left(6-3 x^{4} y ight)\left(6+3 x^{4} y ight) \ ext { (18) } & (3 x-2+y)(3 x-2-y)\end{array} \ ext { (9) }\left(1-a^{4} ight)\left(1-a^{2} ight)\left(1+a^{2} ight) & \end{array} $$

Question

a) Expand and simplify: $$ \begin{array}{llll}	ext { (1) } & (x+7)(x-7) & 	ext { (2) } & (x-3)(x+3) \ 	ext { (4) }(9 x+4)(9 x-4) & 	ext { (5) } & (3 x-2 y)(3 x+2 y) & (2 x-1)(2 x+1) \ \begin{array}{ll}	ext { (7) } & \left(6-3 x^{4} yight)\left(6+3 x^{4} yight) \ 	ext { (18) } & (3 x-2+y)(3 x-2-y)\end{array} \ 	ext { (9) }\left(1-a^{4}ight)\left(1-a^{2}ight)\left(1+a^{2}ight) & \end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's expand and simplify each of the given expressions step by step.

### (1) $$(x + 7)(x - 7)$$

This is a difference of squares:

$$
(x + 7)(x - 7) = x^2 - 7^2 = x^2 - 49
$$

**Simplified Form:** $$x^2 - 49$$

---

### (2) $$(x - 3)(x + 3)$$

Another difference of squares:

$$
(x - 3)(x + 3) = x^2 - 3^2 = x^2 - 9
$$

**Simplified Form:** $$x^2 - 9$$

---

### (4) $$(9x + 4)(9x - 4)$$

Yet another difference of squares:

$$
(9x + 4)(9x - 4) = (9x)^2 - 4^2 = 81x^2 - 16
$$

**Simplified Form:** $$81x^2 - 16$$

---

### (5) $$(3x - 2y)(3x + 2y)$$

This follows the difference of squares pattern where $$a = 3x$$ and $$b = 2y$$:

$$
(3x - 2y)(3x + 2y) = (3x)^2 - (2y)^2 = 9x^2 - 4y^2
$$

**Simplified Form:** $$9x^2 - 4y^2$$

---

### $$(2x - 1)(2x + 1)$$

Applying the difference of squares formula with $$a = 2x$$ and $$b = 1$$:

$$
(2x - 1)(2x + 1) = (2x)^2 - 1^2 = 4x^2 - 1
$$

**Simplified Form:** $$4x^2 - 1$$

---

### (7) $$(6 - 3x^4 y)(6 + 3x^4 y)$$

This is a difference of squares where $$a = 6$$ and $$b = 3x^4 y$$:

$$
(6 - 3x^4 y)(6 + 3x^4 y) = 6^2 - (3x^4 y)^2 = 36 - 9x^8 y^2
$$

**Simplified Form:** $$36 - 9x^8 y^2$$

---

### (18) $$(3x - 2 + y)(3x - 2 - y)$$

This is a difference of squares with $$a = 3x - 2$$ and $$b = y$$:

$$
(3x - 2 + y)(3x - 2 - y) = (3x - 2)^2 - y^2
$$

First, expand $$(3x - 2)^2$$:

$$
(3x - 2)^2 = 9x^2 - 12x + 4
$$

Now, subtract $$y^2$$:

$$
9x^2 - 12x + 4 - y^2
$$

**Simplified Form:** $$9x^2 - 12x + 4 - y^2$$

---

### (9) $$(1 - a^4)(1 - a^2)(1 + a^2)$$

Let's simplify step by step:

1. **Multiply $$(1 - a^2)$$ and $$(1 + a^2)$$:**

$$
(1 - a^2)(1 + a^2) = 1 - a^4
$$

2. **Now, multiply by $$(1 - a^4)$$:**

$$
(1 - a^4)(1 - a^4) = (1 - a^4)^2 = 1 - 2a^4 + a^8
$$

**Simplified Form:** $$1 - 2a^4 + a^8$$

---

## Summary of Simplified Expressions

1. $$(x + 7)(x - 7) = x^2 - 49$$
2. $$(x - 3)(x + 3) = x^2 - 9$$
4. $$(9x + 4)(9x - 4) = 81x^2 - 16$$
5. $$(3x - 2y)(3x + 2y) = 9x^2 - 4y^2$$
6. $$(2x - 1)(2x + 1) = 4x^2 - 1$$
7. $$(6 - 3x^4 y)(6 + 3x^4 y) = 36 - 9x^8 y^2$$
18. $$(3x - 2 + y)(3x - 2 - y) = 9x^2 - 12x + 4 - y^2$$
9. $$(1 - a^4)(1 - a^2)(1 + a^2) = 1 - 2a^4 + a^8$$

If you have any further questions or need additional explanations, feel free to ask!
Here are the simplified forms of the given expressions: 1. $$(x + 7)(x - 7) = x^2 - 49$$ 2. $$(x - 3)(x + 3) = x^2 - 9$$ 4. $$(9x + 4)(9x - 4) = 81x^2 - 16$$ 5. $$(3x - 2y)(3x + 2y) = 9x^2 - 4y^2$$ 6. $$(2x - 1)(2x + 1) = 4x^2 - 1$$ 7. $$(6 - 3x^4 y)(6 + 3x^4 y) = 36 - 9x^8 y^2$$ 18. $$(3x - 2 + y)(3x - 2 - y) = 9x^2 - 12x + 4 - y^2$$ 9. $$(1 - a^4)(1 - a^2)(1 + a^2) = 1 - 2a^4 + a^8$$ If you have any more questions or need further assistance, feel free to ask!

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